Prove that if x is orthogonal to each of v1 V2 vp then x is

Prove that if x is orthogonal to each of v_1, V_2,..., v_p. then x is orthogonal to any vector in Span{v_1, v_2,...,v_p}.

Solution

given \'x\' is orthogonal to vectors v1,v2,....vp

so x.v1 =0 and x.v2=0 ..............x.vp=0

span {v1,v2,...........vp} = av1 +bv2 +................+nvp =0

here a,b, n are any numbers

here v1,v2 , vp deosn\'t change

so x is orthogonal to v1,v2....vp